On partial differential operators which annihilate the roots of the universal equation of degree k

The aim of this paper is to study in details the regular holonomic $D-$module introduced in cite{[B.19]} whose local solutions outside the polar hyper-surface ${Delta(sigma).sigma_k = 0 }$ are given by the local system generated by the local branches of the multivalued function which is the root of the universal degree $k$ equation $z^k + sum_{h=1}^k (-1)^h.sigma_h.z^{k-h} = 0 $. Note that it is surprising that this regular holonomic $D-$module is given by the quotient of $D$ by a left ideal which has very simple explicit generators despite the fact it necessary encodes the analogous systems for any root of the universal degree $l$ equation for each $l leq k$. Our main result is to relate this $D-$module with the minimal extension of the irreducible local system associated to the difference of two branches of the multivalued function defined above. Then we obtain again a very simple explicit description of this minimal extension in term of the generators of its left ideal in the Weyl algebra. As an application we show how these results allow to compute the Taylor expansion of the root near $-1$ of the equation $z^k + sum_{h=-1}^k (-1)^h.sigma_h.z^{k-h} - (-1)^k = 0 $.